Chapter 2
Section 2.1
Are You Prepared for This Section?
P1. The additive inverse of 3 is −3 because
3 + (−3) = 0. The sum of a real number and its
additive inverse is 0.
P2. The multiplicative inverse of − 4 is − 3
3
4
4
3
because −

Answers may vary. Possible answer: The
Subtraction Property of Equality states that for
real numbers a, b, and c, if a = b, then
a − c = b − c. The Division Property of Equality
states that for real numbers a, b, and c, where
c ≠ 0, if a = b, then

105 p = 5250
105
p = 50
The solution of the equation is 50, or the solution
set is {50}.
c − 0.25c = 120
1c − 0.25c = 120
0.75c = 120
100 ⋅ 0.75c = 100 ⋅120
75c = 12,000
75c = 12,000
75
c = 160
The solution of the equation is 160, or the
solution set is {160}.
0.36 y − 0.5 = 0.16 y + 0.3
0.20 y − 0.5 = 0.3
0.2 y = 0.8
10(0.2 y) = 10(0.8)
2y=8
y=4
The solution to the equation is 4, or the solution
set is {4}.
0.12x + 0.05(5000 − x) = 460
0.12x + 250 − 0.05x =
460 0.07x + 250 = 460
0.07x = 210
100(0.07x) = 100(210)
7x = 21,000
x = 3000
The solution to the equation is 3000, or the
solution set is {3000}.
A conditional equation is an equation that is true
for some values of the variable and false for
other values of the variable.
A(n) contradiction is an equation that is false for
every value of the variable. A(n) identity is an
equation that is satisfied for all values of the
variable for which both sides of the equation are
defined.

1, 283, 750 = 25x − 1,872,500 + 1, 031,
250 1, 283, 750 = 25x − 841, 250
2,125, 000 = 25x
85, 000 = x
The adjusted gross income of you and your spouse was $85,000.
Answers may vary. One possibility: The statement 24 = 2 indicates that the original equation is a contradiction.
The solution set is ∅ or { }.
If the last line of the solution of an equation is a true statement, then the solution is all real numbers. If the last
line of the solution of an equation is a false statement, then the solution is the empty set.
Section 2.4
Are You Prepared for This Section?
P1. 2L + 2W for L = 7 and W = 5: 2(7) + 2(5) = 14 + 10 = 24
P2. In 0.5873, the number 8 is in the hundredths place. The number to the right of 8 is 7. Since 7 is greater than or
equal to 5, round 0.5873 to 0.59.
2.4 Quick Checks
A formula is an equation that describes how two or more variables are related.
F=

2. The difference of 7 and 4 is represented
mathematically as 7 − 4.
3. The quotient of 25 and 3 is represented
mathematically as 25 .
3
4. The product of −2 and 6 is represented
mathematically as −2 ⋅ 6.
5. Twice a less 2 is represented mathematically as
2a − 2.
6. Five times the difference of m and 6 is
represented mathematically as 5(m − 6).

In mathematics, English statements can be
represented symbolically as equations.
“The product of 3 and y is equal to 21” is
represented mathematically as 3y = 21.
“The difference of x and 10 equals the quotient
of x and 2” is represented mathematically as

x

.

2
“Three times the sum of n and 2 is 15” is
represented mathematically as 3(n + 2) = 15.

“The sum of three times n and 2 is 15” is
represented mathematically as 3n + 2 = 15.
Letting variables represent unknown quantities
and then expressing relationships among the
variables in the form of equations is called
mathematical modeling.
Find how much each person paid for the pizza. Let
s be the amount that Sean pays. Then Connor
pays
s+

Find the amount invested in each type of
investment. Let s be the amount invested in
stocks. Then the amount invested in bonds is 2s.
The total amount invested is $18,000.
s + 2s = 18,000
3s = 18,000
s = 6000
2s = 2(6000) = 12,000
The amount invested in stocks is $6000, and the
amount invested in bonds is $12,000.
Find the number of miles for which the rental costs
are the same. Let m be the number of miles
driven. Renting from E-Z Rental would cost
30 + 0.15m. Renting from Do It Yourself Rental
would cost
15 + 0.25m.
30 + 0.15m = 15 + 0.25m
30 = 15 + 0.10m
15 = 0.1m

Chapter 2: Equations and Inequalities in One Variable

50 less than half of a number:

1

x − 50

2

the sum of twice a number and 45: 2x + 45
Let r be the number of runs scored by the Tampa
Bay Rays. Then r − 3 is the number of runs
scored by the Toronto Blue Jays.
Let R be the amount Ralph has. Then 3R + 0.25
is the amount Beryl has.
Let j be the amount Juan will get. Then 1500 −
j is the amount Emilio will get.
Let p be the number of paid tickets. Then
12,765 − p is the number of special promotion
tickets.
43 + x = −72

10 ⋅ 15 = 10 ⋅ 0.1m
150 = m
The costs are the same if 150 miles are driven.
Find the number of hours for which the charges
are the same. Let h be the number of hours. A
repair call from Carl’s Appliance Repair Shop
would cost 69.99 + 30h. A repair call from
Terry’s Appliance Repair Shop would cost
54.99 + 40h.
69.99 + 30h = 54.99 + 40h
69.99 = 54.99 + 10h
15 = 10h
1.5 = h
The repair charges are the same for 1.5 hours.

49 = 2x − 3

x

− 15 = 30

−6
2(x + 5) = x + 7
(a) Let x be the number of Facebook users in
the United States and Canada, in millions.
Then x + 98 is the number of Facebook
users in Europe, in millions.
The total number of Facebook users in the
United States, Canada, and Europe is 532
million, so x + (x + 98) = 532.
(a) Let m be the number of magnolia trees.
Then the number of oak trees is 2m − 10.

2.5 Exercises
a number increased by 32.3: x + 32.3
the product of −2 and a number: −2 ⋅ x or −2x
double a number: 2x
8 less than a number: x − 8
the quotient of −14 and a number:

Is $3500 $3000 less than $6500? Yes. Is the sum
of $6500 and $3500 equal to $10,000? Yes. Sean
will receive $6500 and George will receive
$3500.
Find the amount invested in each type of
investment. Let x be the amount invested in
stocks. Then

Find the price of the desk. Let d be the price of
the desk. Then the chair cost

1

d. The total
4
price for the desk and chair is $336.25.

d+

1
4

d = 336.25
d = 336.25

= 269
1

Is 269 plus

(269) equal to 336.25? Yes.

The price of the desk is $269.
Find the amount each person receives. Let x be the
amount Sean receives. Then x − 3000 is the
amount George receives. The total is 10,000.

Find the number of floors in each tower. If x is the
number of floors in the Burj Khalifa, then the
number of floors in the Shanghai Tower is
− 35.
+ (x − 35) = 291
2x − 35 = 291
2x = 326
= 163
If x = 163, then x − 35 = 128. Is 128 equal to
35 less than 163? Yes. Do 163 and 128 sum to
291? Yes. The Burj Khalifa has 163 floors and
the Shanghai Tower has 128 floors.

2

3

⋅ 40, 000

5

= 24, 000
x = 16, 000
Is 16,000

2

of 24,000? Yes. Is the sum of 3

24,000 and 16,000 equal to 40,000? Yes. Jack
and Diane should invest $24,000 in stocks and
$16,000 in bonds.
Find the cost of the paperback book. Let x be the
cost of the hardback book. Then x − 12.50 is the
cost of the paperback. The sum of the prices is
37.40.
+ ( x − 12.50) = 37.40
2 x − 12.50 = 37.40
2 x = 49.90
x = 24.95
− 12.50 = 12.45
Is $12.45 $12.50 less than $24.95? Yes. Do
$24.95 and $12.45 sum to $37.40? Yes. The
paperback book costs $12.45.
Find the cost of the bathing suit. Let x be the cost
of the shorts. Then x + 8 is the cost of the
bathing suit and x − 2 is the cost of the T-shirt.
Their costs total $60.
+ ( x + 8) + ( x − 2) = 60
3 x + 6 = 60
3 x = 54
= 18
If x = 18, then x + 8 = 26 and x − 2 = 16. Is $26
$8 more than $18? Yes. Is $16 $2 less than $18?
Yes. Do $18, $26, and $16 sum to $60? Yes. The
bathing suit costs $26.

Find the number of monthly minutes for which
the cost is the same. Let x be the number of
minutes. Then Company A charges
12 + 0.1x and Company B charges 0.15x.
12 + 0.1x = 0.15x
12 = 0.05x
240 = x
If 240 minutes are used, then Company A
charges 12 + 0.1(240) = 12 + 24 = $36 and
Company B charges 0.15(240) = $36, so the
charges are the same. The cost is the same when
240 minutes are used.
Find the number of vacuums sold for which the
jobs pay the same. Let x be the number of
vacuums sold. The first job pays 2000 + 50x and
the second job pays 1200 + 60x.
2000 + 50 x = 1200 + 60x
2000 = 1200 + 10x
800 = 10x
80 = x
If 80 vacuums are sold, the first job pays
2000 + 50(80) = 2000 + 4000 = $6000 and the
second job pays
1200 + 60(80) = 1200 + 4800 = $6000, so they
are the same. The jobs pay the same for 80
vacuums sold.
Find the cost of each item. Let x be the cost of
the lantern. Then x + 30 is the cost of the
cookware and x + 34 is the cost of the cook
stove. The total cost is $199.
+ ( x + 30) + ( x + 34) = 199
3 x + 64 = 199
3 x = 135
x = 45
If x = 45, then x + 30 is 75 and x + 34 is 79. Is
$75 $30 more than $45? Yes. Is $79 $34 more
than $45? Yes. Is the sum of $45, $75, and $79
equal to $199? Yes. The lantern cost $45, the
cookware cost $75, and the cook stove cost $79.
Find the score Brooke needs on the final exam.
There are 5 tests, plus the final exam which is
worth two tests, for a total of 7 tests. The
average must be 80.
80 + 83 + 71 + 61 + 95 + 2x =

Find the wholesale price of the gasoline. Let p
represent the wholesale price of the gasoline.
The selling price is the sum of the wholesale
price and the markup.
p + 0 .10 p = 2 .64
1 .10 p = 2 .64

= 2 .4
The gas station pays $2.40 per gallon for the
gasoline.
Find the original price of the recliners. Let p
represent the original price. The sale price is the
original price less the 25% discount.
p − 0.25 p = 494.25
0.75 p = 494.25
0.75 p = 494.25

0.75
0.75
p = 659
The original price of the recliners was $659.
Find the value of Albert’s house one year ago. Let
x represent the value one year ago. The value
now is the value a year ago minus the 2% loss.
− 0. 02 x = 148, 000
0 .98 x = 148, 000

Let x be the value of the home one year ago.
Then 0.04x is the increase in value.
+ 0.04 x = 208, 000
1.04 x = 208, 000
= 200, 000
The home was valued at $200,000 one year ago.
Let x be the sale price.
= 28,000 − 0.15(28,000)
= 28,000 − 4200
= 23,800
The sale price of the truck is $23,800.
Let x be the amount the bookstore paid. Then
0.30x is the markup amount.
+ 0.30 x = 117
1.30 x = 117
x ≈ 90
The bookstore paid $90 for the book.
Let x be the price of the vacation package before
the sale. Then 0.13x is the discount amount.
− 0 .13 x = 1007
0 .87 x = 1007
≈ 1157
The vacation package was priced at $1157
before the 13%-off sale.
Let p be the number of Democrats on the
committee. Then there are 2p Republicans on the
committee. Also, 0.3(2p) Republicans and 0.2p
Democrats voted in favor of a bill.
0.3(2 p ) + 0.2 p = 8
0.6 p + 0.2 p = 8
0.8 p = 8
p = 10
2p = 20
There are 20 Republicans and 10 Democrats or
30 people on the committee.
Let f be the number of full-page ads. Then 2f is
the number of half-page ads. He receives
0.08(450) for each full-page ad and 0.05(300)
for each half-page ad.
0.08(450) f + 0.05(300)(2 f ) = 5610
36 f + 30 f = 5610
66 f = 5610
f = 85
2 f = 170
Mario brought in 85 full-page ads and 170 halfpage ads.

96

ISM: Elementary & Intermediate Algebra

Let x be the number of students in the class.
Then 0.15x of the students received an A.
0.15 x = 6
x = 40
There were 40 students in his astronomy class.
Let x be the number that have never married.
x = 0.17(125)
x = 21.25
Therefore, 21 million females aged 18 years or
older had never married.
Let x = percent of population that held an
associate’s degree in 2004.
13, 244
=
⋅ 100 % ≈ 7 .
1% 186, 534
Let y = percent of population that held an
associate’s degree in 2015.
20, 867
=
⋅ 100 % ≈ 9 .
8% 212, 132
The percent in 2004 was 7.1% and in 2015 was
9.8%.
The amount of change is 2825 − 2500 = 325.
325 =
0.13

157 .37
The percent increase is 2.00%.
Answers may vary. One possibility: If an item
sells for p dollars, the tax on the item is 0.06p, so
the total purchase price is p + 0.06p = 1.06p.
Reducing an item by 10% and then reducing that
same item by 20% is not the same as reducing
the item by 30%. Reducing an item whose
original cost is x by 10% produces a new price of
0.9x. Reducing that price by 20% yields a new
price of (0.9x)(0.8) = 0.72x.

2.7 Quick Checks
Complementary angles are angles whose
measures have a sum of 90 degrees.
This is a complementary angle problem. We are
looking for the measures of two angles whose
sum is 90°. Let x represent the measure of the
smaller angle. Then x + 12 represents the
measure of the larger angle.
+ ( x + 12) = 90
2 x + 12 = 90
2 x = 78
x = 39
x + 12 = 39 + 12 = 51
The two complementary angles measure 39° and
51°.
This is a supplementary angle problem. We are
looking for the measures of two angles whose
sum is 180°. Let x represent the measure of the
smaller angle. Then 2x − 30 represents the
measure of the larger angle.
+ (2 x − 30) = 180
3 x − 30 = 180
3 x = 210
x = 70
2x − 30 = 2(70) − 30 = 140 − 30 = 110
The two supplementary angles measure 70° and
110°.
The sum of the measures of the angles of a
triangle is 180 angles.
This is an “angles of a triangle” problem. We
know that the sum of the measures of the interior
angles of a triangle is 180°. Let x represent the

x=

x = 105

1

True
False; the perimeter of a rectangle can be found
by adding twice the length of the rectangle to
twice the width of the rectangle.
This is a perimeter problem. We want to find the
width and length of a garden. Let w represent the
width of the garden. Then 2w represents the
length of the garden. We know that the perimeter
is 9 feet, and that the formula for the perimeter
of a rectangle is P = 2l + 2w.
2(2 w) + 2 w =
94w+2w
=96w=9
w=

9
6

w=

3
2

2w=2

3

=3

2
The width of the garden is

3

= 1.5 feet and the 2

length is 3 feet.
This problem is about the surface area of a
rectangular box. The formula for the surface area
of a rectangular box is SA = 2lw + 2lh + 2hw,
where l is the length of the box, w is the width of
the box, and h is the height of the box. We are
given the surface area, the length, and the width
of the rectangular box.

= 10h
=h
The height of the box is 5 feet.
False; when using d = rt to calculate the distance
traveled, it is necessary to travel at a constant
speed.
This is a uniform motion problem. We want to
know the average speed of each biker. Let r
represent the average speed of Luis. Then r + 5
represents Mariko’s average speed. Each biker
rides for 3 hours.
⋅

Rate
Luis
Mariko

Time = Distance

r

3

3r

r+5

3

3(r + 5)

Since the bikers are 63 miles apart after
3 hours, the sum of the distances they biked is
63.
3r + 3( r + 5) = 63
3r + 3r + 15 = 63
6 r + 15 = 63
6 r = 48
=8
r + 5 = 8 + 5 = 13
Luis’ average speed was 8 miles per hour and
Mariko’s average speed was 13 miles per hour.
This is a uniform motion problem. We want to
know how long it takes to catch up with Tanya,
and how far you are from your house when you
catch her. Let x represent the amount of time you
drive before catching Tanya. Then x + 2
represents the number of hours that Tanya runs
before you catch her.
Rate
Tanya
You

⋅

Time = Distance

8

x+2

8(x + 2)

40

x

40x

When you catch up with Tanya, the distances the
two of you have traveled is the same.

98

40 x = 40

= 20

2
You and Tanya are 20 miles from your house
when you catch up to her.
2.7 Exercises
We are looking for the measures of two angles
whose sum is 180°. Let x represent the measure
of the second angle. Then 4x represents the
measure of the first angle.
4 x + x = 180
5 x = 180
= 36
4x = 4(36) = 144
The angles measure 36° and 144°.
We are looking for the measures of two angles
whose sum is 90°. Let x represent the measure of
the second angle. Then x − 25 represents the
measure of the second angle.
( x − 25) + x = 90
2 x − 25 = 90
2 x = 115
= 57 .5
x − 25 = 57.5 − 25 = 32.5
The angles measure 57.5° and 32.5°.
18. x + (2 x + 1) +

We want to find the measures of the angles of a triangle. The measures of the angles of a triangle sum to 180°. Let
be the measure of the first angle. Then x + 2 and x + 4 are the measures of the next two angles, respectively.
+ ( x + 2) + ( x + 4) = 180
3 x + 6 = 180
3 x = 174
x = 58
If x = 58, then x + 2 = 60, and x + 4 = 62. Do 58, 60, and 62 sum to 180? Yes. The measures of the angles are 58°,
60°, and 62°.
We want to find the dimensions of a rectangle. The perimeter of a rectangle is twice the length plus twice the
width. Let l be the length of the rectangle. Then the width is l − 10. The perimeter is 56 meters.
2l + 2( l − 10) = 56
2l + 2l − 20 = 56
4l = 76
l = 19
If l = 19, then 19 − 10 = 9. Is the sum of twice 19 and twice 9 equal to 56? Yes. The length of the rectangle is
19 meters and the width is 9 meters.
We want to find the dimensions of the rectangular plot. Let w be the width (shorter dimension) of the original plot.
Then the length of the plot is 3w and the sides of the smaller square field are all w. There are 9 lengths of w that
need fencing (3 for each long side, 1 for each end, and one to divide) and 279 meters of fencing was used.
9 w = 279
w = 31
If w = 31, then 3w = 93. Would it require 279 meters of fencing to enclose a plot that is 31 feet by 93 feet and
divide it into two parcels? Yes. The plot is 31 meters by 93 meters.
We want to find the dimensions of the parallelogram. The perimeter is 120 inches. Let x be the length of the
shorter side. Then x + 10 is the length of the longer side.
P = 2l + 2w 120
= 2( x + 10) + 2x
120 = 2 x + 20 + 2x
120 = 4 x + 20 100
= 4x
25 = x
If x = 25, then x + 10 = 35. Is twice 25 plus twice 35 equal to 120? Yes. The parallelogram is 25 inches by
35 inches.
(a) Let t be the time since the trains left Albuquerque. The distance traveled by the faster train is 72t.
The distance traveled by the slower train is 66t.
The difference in distance is 72t − 66t.
An equation is 72t − 66t = 45.
30.
beginning of trip
rest of the trip
total

We want to find the sides of a triangle. The
perimeter of a triangle is the sum of the lengths
of the legs. Let x be the length of each of the
congruent legs. The base is 17 centimeters and
the perimeter is 95 centimeters.
2 x + 17 = 95
2 x = 78
x = 39
Is the sum of 39, 39, and 17 equal to 95? Yes.
Each leg is 39 centimeters.

2 s + ( s + 5) = 41
3 s + 5 = 41
3 s = 36
= 12
s + 5 = 12 + 5 = 17
The speed of the boat to the islands was 12 miles
per hour and the speed on the return trip was 17
miles per hour.
Let t be the time it takes them to get to school.

To solve an inequality means to find the set of all
replacement values of the variable for which
the statement is true.
3<8
3+7<8+7
10 < 15
This illustrates the Addition Property of
Inequality.
n−2>1
n−2+2>1+2
n>3
The solution set is {n|n > 3} or (3, ∞).
1

We want to know the maximum number of boxes
of supplies that the worker can move on the
elevator. Let b represent the number of boxes.
Then the weight of the boxes is 91b, and the
weight of the worker and the boxes is
180 + 91b. This weight cannot be more than
2000 pounds.
180 + 91b ≤ 2000
91b ≤ 1820
91b ≤ 1820
91
b ≤ 20
The number of boxes must be less than or equal
to 20, so the maximum number of boxes the
worker can move in the elevator is 20.

(−2, ∞)
Adding 2 to each side does not change the
direction of the inequality symbol. The symbol
remains >. We used the Addition Principle of
Inequality.
Multiplying each side by

does not change the direction of the inequality
symbol. The symbol remains ≥. We used the
Multiplication Principle of Inequality.
Subtracting 1 or adding −1 to each side does not
change the direction of the inequality symbol.
The symbol remains ≤. We used the Addition
Principle of Inequality.
Dividing each side by −4 or multiplying each
side by −

0

8

x≤6

, a negative number, reverses the
4
direction of the inequality symbol. The symbol
becomes >. We used the Multiplication Principle
of Inequality.

The direction of the inequality sign is reversed
when each side is multiplied or divided by a
negative number.
The notation [−7, −∞) is incorrect for several
reasons. The interval should be written in order
from smaller to larger. Since we want values
greater than −7, the interval should go to positive
infinity. The left symbol should be a parenthesis.
The correct notation is (−7, ∞).
Chapter 2 Review
3x + 2 = 7; x = 5
3(5) + 2 7
15 + 2 7
17 = 7 False
No, x = 5 is not a solution to the equation.
5m − 1 = 17; m = 4
5(4) − 1 17
20 − 1 17
19 = 17 False
No, m = 4 is not a solution to the equation.

numbers 12, 13, 14 consecutive integers? Yes.
Do they sum to 39? Yes. The integers are 12, 13,
and 14.

the difference between a number and 6: x
−6
eight subtracted from a number: x − 8

We want to find how much each will receive.
Let j be the amount received by Juan. Then j
− 2000 is the amount received by Roberto. j
+ ( j − 2000) = 20,000
2 j − 2000 = 20,000
2 j = 22,000
j = 11,000
If j = 11,000, then j − 2000 = 9000. Do 11,000
and 9000 differ by 2000? Yes. Do 11,000 and
9000 sum to 20,000? Yes. Juan will receive
$11,000 and Roberto will receive $9000.

the product of −8 and a number: −8x
the quotient of a number and 10:

x
10

twice the sum of 6 and a number: 2(6 + x)
four times the difference of 5 and a number:
4(5 − x)
6 + x = 2x + 5

We want to find the number of miles for which
the cost will be the same. Let x be the number of
miles driven. ABC-Rental charges 30 + 0.15x
and U-Do-It Rental charges
15 + 0.3x.
30 + 0.15x = 15 + 0.3x
30 = 15 + 0.15x
15 = 0.15x 100
=x
ABC-Rental’s cost will be
30 + 0.15(100) = 30 + 15 = $45 and
U-Do-It-Rental’s cost will be
15 + 0.3(100) = 15 + 30 = $45, and they are the
same. The cost will be the same for 100 miles.

We want to find supplementary angles. The
measures of supplementary angles sum to 180°.
Let x be the measure of the second angle. Then
2x − 60 is the measure of the first angle.
+ (2x − 60) = 180
3x − 60 = 180
3x = 240
x = 80
If x = 80, then
2x − 60 = 2(80) − 60 = 160 − 60 = 100. The
measures of the angles are 80° and 100°.
We want to find the measures of the angles of the
triangle. The measures of the angles of a
triangle sum to 180°. Let x be the measure of the
first angle. Then 2x is the measure of the second
and 2x + 30 is the measure of the third.
+ (2x ) + (2x + 30) = 180
5x + 30 = 180
5x = 150
x = 30
If x = 30, then 2x = 60, and
2x + 30 = 60 + 30 = 90. The measures of the
angles are 30°, 60°, and 90°.
We want to find the measures of the angles of the
triangle. The measures of the angles of a
triangle sum to 180°. Let x be the measure of the
first angle. Then x − 5 is the measure of the
second angle and 2(x − 5) − 5 is the measure of
the third angle.
+ (x − 5) + 2(x − 5) − 5 = 180
x + x − 5 + 2x − 10 − 5 = 180
4x − 20 = 180
4x = 200
x = 50
If x = 50, then x − 5 = 45 and
2(x − 5) − 5 = 2(45) − 5 = 85. The measures
of the angles are 50°, 45°, and 85°.
We want to find the dimensions of the rectangle.
Let w be the width of the rectangle. Then
2w + 15 is the length of the rectangle. The
perimeter of a rectangle is the sum of twice the
length and twice the width. The perimeter is
78 inches.
2(2w + 15) + 2w = 78
4w + 30 + 2w = 78
6w + 30 = 78
6w = 48
w=8
If w = 8, then 2w + 15 = 2(8) + 15 = 31. The
length is 31 inches and the width is 8 inches.

We want to find the dimensions of the rectangle.
Let w be the width of the rectangle. Then 4w is
the length of the rectangle. The perimeter of a
rectangle is the sum of twice the length and
twice the width. The perimeter is 70 cm.
2(4w) + 2w = 70
8w + 2w = 70
10w = 70
w=7
If w = 7, then 4w = 28. The width of the
rectangle is 7 cm and the length is 28 cm.
(a) We want to find the dimensions of the
rectangular garden. Let l be the length. Then
the width is 2l. The perimeter is twice the
length plus twice the width and is 120 feet.
2l + 2(2l) = 120
2l + 4l = 120
6l = 120
l = 20
If l = 20, then 2l = 40. The garden’s length
is 20 feet and the width is 40 feet.
A = lw = 20(40) = 800
The area of the garden is 800 square feet.

Chapter 2: Equations and Inequalities in One Variable

25t − 18t = 35
7t = 35
=5
They will be 35 miles apart after 5 hours.
Let r be the speed of the faster train.
Rate

The area of a trapezoid is A = h (B + b), where 2
h is the height and the bases are B and b. Let B
be the longer base. Then B − 10 is the shorter
base. The height is 80 feet and the area is
3600 square feet.
(80)[B + (B − 10)] = 3600
40(2B − 10) = 3600
80B − 400 = 3600
80B = 4000
= 50
If B = 50, then B − 10 = 40. The bases are 50
feet and 40 feet.
Let t be the time at which they are 35 miles
apart.
Rate

⋅ Time

= Distance

slow

18

t

18t

fast

25

t

25t

0

4

m<2
0

2

m ≥ −5
–5

0

0<n
0

−3 ≤ n
–3

0

x < −4
(−∞, −4)
x≥7
[7, ∞)
[2, ∞)

The difference of the distances is 35, since they
are traveling in the same direction.

− 2x + 2x + 14 = − 2x + 2x + 14
14 = 14
This is a true statement. The equation is an
identity. The solution set is the set of all real
numbers.

= lwh
wh

wh

=l

wh
l=

m + (m + 2) + (m − 14) = 60

V

3m − 12 = 60

; V = 540, w = 6, h = 10

wh
l = 540 = 540 = 9
6(10)
60
The length is 9 inches.
10. (a)

= 15
If n = 15, then n + 1 = 16 and n + 2 = 17. Do 15,
16, and 17 sum to 48? Yes. Are 15, 16, and 17
consecutive integers? Yes. The integers are 15,
16, and 17.
We need to find the lengths of the three sides.
Let m be the length of the middle side. Then the
length of the longest side is m + 2, and the length
of the shortest side is m − 14. The perimeter, or
the sum of the three sides, is 60.

3m = 72
m = 24
If m = 24, then m + 2 = 26, and m − 14 = 10.
Do 24, 26, and 10 sum to 60? Yes. The lengths
of the sides are 10 inches, 24 inches, and
26 inches.
Let t be the time at which they are 350 miles
apart.
Rate

The sum of their distances is 350 since they are
traveling in opposite directions.
40t + 60t = 350
100t = 350
= 3.5
They will be 350 miles apart in 3.5 hours.
Let x be the length of the shorter piece. Then
3x + 1 is the length of the longer piece. The sum
of the lengths of the pieces is 21.
+ (3x + 1) = 21
4x + 1 = 21
4x = 20
x=5
3x + 1 = 3(5) + 1 = 15 + 1 = 16
The shorter piece is 5 feet and the longer piece is
16 feet.
Let x be the original price of the backpack. Then
the discount amount was 0.20x
x − 0.20x = 28.80
0.80x = 28.80